Non myopic strategy
Truth or Lie?
Scoring Rules
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One important feature of market scoring rules is that
they are myopic strategy proof.
That means that it is optimal for a trader to report his
true belief about the likelihood of an event.
But ignore the impact of his report on the profit he
might get from future trades.
Hence, Bluffing first and telling the truth might be
better than telling the truth!
What’s in today class?
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Simple model (2 partially informed traders in
single information market).
Extend results to more complicated markets
with multiple traders and signals.
New scoring rule which reduces the
opportunity for bluffing strategies.
Introduction
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Information market gather data on future events. An
informed trader can use his private information to
recognize inaccuracies and make profit.
Those trades influence the trading price.
Those price provide signals to others about the
future event which causes them to adjust their belief
about the true value of their security.
Example
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Say in the 2012 US election campaign.
When market price showed probability of 50% that
either Obama or Romney will win.
And, I thought Romney will win.
Day before the election I checked and saw that there
is 70% that Obama will win. I tell myself if the
markets thinks Obama is going to win I should
change my mind and join the market opinion!
Ideals…
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Ideally, this will lead to a situation in which all
traders reach common consensus that
reflects all available information.
Hence the conclusion that prediction markets
rely on traders adjusting their beliefs in
response to other traders trade (the market
price)
There is your drawback
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Traders can mislead other traders about the
value of their security and than profit from
their mistake in some later trade!
This can also cause traders to be cautious
about making inferences from market prices,
thus damaging the data aggregation of the
market.
2-players and 2 signals
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Need to predict future event E.
Two players: P1, P2. each has private info about
E.(say x1,x2)
Assume traders share the prior probability
distribution.
More assumptions
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The two signals are independent.
X1 is 0,1 with prob 0.5,0.5
X2 is 0,1 with prob q, 1-q (0<q<1)
E(E/x1x2=00) = p00, E(E/x1x2=10) = p10 and so on.
The model can be fully specified:
Trades
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Trade in security F base on prediction of
event E is done using a market scoring rule.
Players make sequence of market moves.
In each move the player announces the a
probability Pi for the event E.
In the paper they look on logarithmic scoring
rule.
Let’s play
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Suppose P1 saw x1 = 1. and he is myopic.
She will calculate the probability of the event
as:
r1 = q*P11 + (1-q)*P10
and she will therefore bid r1,
(If x1=0 she will bid r0=q*P01+(1-q)*P00)
Next turn
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P2 doesn’t know what x1 is but he can infer it
from the bid P1 made, i.e., r0 or r1.
P2 infers the x1 and knows x2 so he can post
the best estimate of the conditional
provability of E.
P2 will bid p00 or p01 or p10 or p11.
No one else will want to move.
Good market
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All information will aggregate in just 2 steps.
Both players would make profit in
expectation in the market.
Bluffing
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Same thing but player 1 decides to bluff.
Which means he sees x1=1 but moves to r0.
P2 infers P1 saw x1=0 and infers the
probability knowing x2. (p00 or p01)
P1 sees the choice of P2, learns x2 and
know x1 and moves to the best probability.
Bluff or tell the truth?
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P1 lies because she has greater profit when
doing so vs. playing myopically.
If P1 will benefit more from telling the truth,
P2 have no reason to lie since P1 will not
play again.
So bluffing can start only if P1 calculates that
bluffing is better than telling the truth.
Bluffing by Player 1 (P2 is myopic)
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If bluffing is more profitable then P1 will bluff
with some probability s. P2 can analyze P1’s
profit in different scenarios and decide that
P1 is bluffing.
If P2 knows, his best response based on s,
X2 and the price r0 or r1 has published can
be calculated and he too can decide if to use
it or to bluff.
Bluffing is Profitable
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Let’s prove that player 1 has incentive to bluff
which on information market with logarithmic
scoring rule.
General Informativeness condition
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We limit ourselves to the case in which every
agent has something to tell us about the
world, no matter what the other agents tell
us.
Therefore we have to learn what each agent
knows to make the best bid.
Let’s prove
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2 players, 2 signals.
Event prob. conditioned upon P1 seeing i
and P2 seeing j is Pij
Player 2 has prob. q to see 1.
Player’s 1 myopic bids are:
r1 = q p11 + (1-q)p10
r0 = q p01 + (1-q)p00
Equilibrium strategy profile
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Each player’s strategy is sequentially rational
, which meant that each player wants to
maximize his own profit in expectation from
making trades in the market, given all
information that he knows on the event at
each point of the game.
Lemma 1
Proof
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Assume Ru != R1.
Whenever player 1 will play Ru, player 2 will
deduce player 1 saw 1 and will profit the
remaining surplus.
So, Player one will always earn profit from
only the first move, but by definition of
myopic optimality R1 will yield better profit!
Two strategies
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Myopic: Ps -> R1
Bluff: Ps->R1->R0
Ps->R1 cancels out.
Analyze the profit or loss from the movie from
R1->R0
Scoring & Antropy
Lemma 2 – the profit of bluffing
Proof
Weak PBE strategy porfile
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1) Unique case of general Equilibrium
strategy profile.
2) Strategies are sequentially rational given
their beliefs.
3) Updating the players beliefs is base on
using Bayes’s rule given the strategies.
Theorem 3 – Bluffing is as good as
telling the truth
Proof
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Let (S1,S2) be a weak PBE strategy.
Suppose S1 requires P1 to follow myopic
strategy in the first round.
By lemma 1, P1 will have to bid r1 or r0.
(when P1 sees x1=1 or x1=0 accordingly)
P2 will take into account and move to the
optimal point. (p00, p01, p10, p11)
Proof
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Consider a deviation from this strategy in which
P1 bluffs and corrects P2’s move at the end.
Lemma 2 shows the expected additional score
increase if P1 bluffed:
Proof
Proof
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Inequality is strict when q!=0,1 and p10 !=
P00,P11
Thus, bluffing will be strictly profitable
deviation under this thereom.
Hence, myopic strategy of P1 cannot be part
of an equilibrium profile.
No promised convergence
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We showed that it is always profitable for the
player to bluff and not play deterministically.
There are 2 cases:
Case 1: player 1 plays some strategy
regardless of x1, In this case P2 learnt
nothing about x1 and the theorem always
holds after round 1.
No promised convergence
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P1 plays mixed strategies and moves to Ru which
with prob. T when x1=1 and prob. T` when x1=0
when T and T’ are non zero.
so P2 can not infer exactly what is x1. so P2 assigns
some prob. K to x1 = 1. (K != 0,1)
The conditions in theorem 3 holds so P2 has also
incentive to bluff.
Thus the price cannot converge with certainty after N
finite rounds!
Generalize the results
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Lets look on m players with n signals.
Player 1 move
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Player’s 1 myopic optimal moves:
Player’s 1 decides to bluff:
Homework
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Prove claim 5.
Bluffing is again better
Note about convergence
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As in the 2 players model, convergence is
not guaranteed, because bluffing is better in
each round.
Fight the bluff
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We want to cancel the incentive to bluff.
Let’s reduce the price paid for future trades!
Maybe even cause traders to take the
myopic strategy.
New Scoring Rule:
Truth or lie?
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The myopic strategies hold, since every round we
multiply by const.
But this scoring rule is better on non myopic
strategies.
Di quantifies the degree of aggregation in the
prediction market.
Profit
Payoff gets smaller
Note on delta
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Choosing small delta will speed up
convergence to real price because the
benefit from each additional trade will be
less.
However too rapid drop will cause traders not
to participate because of too small price.
Thank you!